Absolute value of a real number
A quality check, a survey with a margin of error or the change in a price: in every case we want to know how far a value deviates from its reference, whether it is above or below. Absolute value formalises that idea and avoids sign errors that, in a manufacturing tolerance or a statistical report, change the conclusion.
Formal definition by cases
Calculate the numerical value of the following expressions using the definition of modulus: a) $|-12,5|$ b) $|7 - 15|$ c) $|\pi - 4|$ d) $|1 - \sqrt{2}|$
Calculate the numerical value of the following expressions using the definition of modulus:
- a) $|-12,5|$
- b) $|7 - 15|$
- c) $|\pi - 4|$
- d) $|1 - \sqrt{2}|$
The temperature in a city on the Spanish plateau at 6:00 a.m. was $-3\,^\circ C$, while at 3:00 p.m. it rose to $14\,^\circ C$. Use the concept of absolute value to…
The temperature in a city on the Spanish plateau at 6:00 a.m. was $-3\,^\circ C$, while at 3:00 p.m. it rose to $14\,^\circ C$. Use the concept of absolute value to express and calculate the temperature distance (the difference in degrees) between the two measurements.
In the figure you can see the two temperatures placed on the number line of degrees.
- a) Write, using absolute value, the expression that gives the distance between the two temperatures.
- b) Calculate its value and interpret it in the context of the problem.
Equations with absolute value
Find the values of $x$ that satisfy the following equalities: a) $|x| = 8$ b) $|x - 3| = 5$ c) $|x + 2| = 0$
Find the values of $x$ that satisfy the following equalities:
- a) $|x| = 8$
- b) $|x - 3| = 5$
- c) $|x + 2| = 0$
Calculate the values of $x$ that satisfy the following equation: $$|2x + 8| = 14$$
Calculate the values of $x$ that satisfy the following equation:
$$|2x + 8| = 14$$Inequalities with absolute value
Represent on the real number line and write in interval form the values of $x$ that satisfy the following inequality: $$|x - 5| \le 2$$ You can do it on paper and send a…
Represent on the real number line and write in interval form the values of $x$ that satisfy the following inequality:
$$|x - 5| \le 2$$You can do it on paper and send a photo with your answer.
Solve the following absolute value inequality and represent the solution on the real number line. Express the final result using interval notation: $$|x - 4| \leq 3$$
Solve the following absolute value inequality and represent the solution on the real number line. Express the final result using interval notation:
$$|x - 4| \leq 3$$Solve the inequality $|x| > 5$ and represent the solution graphically. Remember that this type of expression gives rise to a union of two infinite rays.
Solve the inequality $|x| > 5$ and represent the solution graphically. Remember that this type of expression gives rise to a union of two infinite rays.
Neighbourhoods and tolerance margins
En una fábrica de componentes electrónicos, una resistencia debe tener un valor nominal de $220\ \Omega$. Por control de calidad, se acepta que el valor real $x$ de la…
En una fábrica de componentes electrónicos, una resistencia debe tener un valor nominal de $220\ \Omega$. Por control de calidad, se acepta que el valor real $x$ de la resistencia varíe respecto al nominal en un máximo de $5\ \Omega$.
En la recta de la figura se marca el valor nominal y un valor real cualquiera $x$, junto con su distancia al valor nominal.
- a) Escribe esta condición utilizando una desigualdad con valor absoluto.
- b) Resuelve la desigualdad y determina el intervalo de valores óhmicos aceptables para estas resistencias.
In a statistical study on the monthly leisure expenditure of young people in a city, it has been determined that the mean is $45$ € with a maximum error of $12$ €. a)…
In a statistical study on the monthly leisure expenditure of young people in a city, it has been determined that the mean is $45$ € with a maximum error of $12$ €.
- a) Express this situation using a neighbourhood $E(a, r)$.
- b) Write the corresponding modulus inequality.
- c) Could a young person spending $32$ € per month be included in this study group? Justify your answer.
Combining conditions and nested neighbourhoods
Find the interval resulting from the following combined condition and represent it graphically: $$|x| > 2 \quad \text{and} \quad |x| \leq 5$$
Find the interval resulting from the following combined condition and represent it graphically:
$$|x| > 2 \quad \text{and} \quad |x| \leq 5$$A financial analyst observes that the price of a cryptocurrency has fluctuated following the rule $|P - 450| < 75$, where $P$ is the price in euros. At the same time,…
A financial analyst observes that the price of a cryptocurrency has fluctuated following the rule $|P - 450| < 75$, where $P$ is the price in euros. At the same time, another source indicates that for security reasons, the investment is only recommended if $|P - 500| > 50$.
Determine the price interval in which both conditions are satisfied to make a safe investment according to these data.